nLab
Klein four-group

Contents

Contents

Idea

The direct product group of the group of order 2 with itself is known as the Klein four group:

/2×/2. \mathbb{Z}/2 \times \mathbb{Z}/2 \,.

Properties

General

Besides the cyclic group /4\mathbb{Z}/4 of order 4, the Klein group /2×/2\mathbb{Z}/2 \times \mathbb{Z}/2 is the only other group of order 4, up to isomorphism.

In particular the Klein group is not itself a cyclic group, and it is in fact the smallest non-trivial group which is not a cyclic group.

ADE-Classification

In the ADE-classification of finite subgroups of SO(3), the Klein four-group is the smallest in the D-series, labeled by D4.

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
D4Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group
SO(2n)SO(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

References

See also

Last revised on February 18, 2019 at 08:03:30. See the history of this page for a list of all contributions to it.